Modern Control Engineering

136 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systems

ab+b +– Ks

E(s) X(s) Y(s)

Z(s) T

1 s

T 1 T 2 s^2 + (T 1 + 2 T 2 )s+ 1

a

a+b

Figure 4–25

Block diagram for

the system shown in

Figure 4–24.

A block diagram for this system is shown in Figure 4–25. The transfer function

Y(s)/E(s)can be obtained as

Under normal circumstances we design the system such that

then

where

Thus, the controller shown in Figure 4–24 is a proportional-plus-integral-plus-derivative

controller (PID controller).

4–5 Thermal Systems

Thermal systems are those that involve the transfer of heat from one substance to

another. Thermal systems may be analyzed in terms of resistance and capacitance,

although the thermal capacitance and thermal resistance may not be represented

accurately as lumped parameters, since they are usually distributed throughout the sub-

stance. For precise analysis, distributed-parameter models must be used. Here, however,

to simplify the analysis we shall assume that a thermal system can be represented by a

lumped-parameter model, that substances that are characterized by resistance to heat

flow have negligible heat capacitance, and that substances that are characterized by heat

capacitance have negligible resistance to heat flow.

Kp=

b

a

T 1 +2T 2

T 1

, Ki=

b

a

1

T 1

, Kd=

b

a

T 2

=Kp+

Ki

s

+Kd s

Y(s)

E(s)

=

b

a

T 1 T 2 s^2 +AT 1 +2T 2 Bs+ 1

T 1 s

`

a

a+b

K

s

T 1 s

T 1 T 2 s^2 +AT 1 +2T 2 Bs+ 1

`  1

Y(s)

E(s)

=

b

a+b

K

s

1 +

a

a+b

K

s

T 1 s

T 1 T 2 s^2 +AT 1 +2T 2 Bs+ 1

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